Fractals are very interesting. There are different ways to describe one, but one way to think of one is that it’s a shape that looks the same no matter what magnification you use. You can double it, triple it, make it 10,876,432 times bigger, and the object still displays (more or less) the same features. The term fractal was coined by Benoît Mandelbrot, and there is an entire subclass of fractals named after him. They are seen in nature (and art, like here) quite a bit. Coastlines are fractal, as are — seriously — some kinds of broccoli.

However, fractals are generally calculated in two dimensions. What’s new here is that the fractal pattern has now been calculated in three dimensions! That is, to say the least, a non-trivial procedure — I used to play with some of the 2D equations many years ago, on my old 512k Fat Mac, with code written in Pascal (yes, with the semicolons and everything) and it was fascinating if very complex.

But the 3D idea has been written up by Daniel White, who, along with others, figured out how to create and render such an incredible object. He even created a "fly-over" video to demonstrate the fractal pattern:

Wow. Even if the math of this makes no sense at all to you, the beauty of this should be apparent.

Which brings up a point: why are mathematical shapes beautiful? What makes them so pleasing to our eyes and brain; why did we evolve an appreciation for such things? I don’t know, and at some point I’ll have to research that a bit — understanding the principles behind this will help me appreciate it even more.

Fractals. That reminds me of an experiment I had to do during my studies. It showed that a fractal had a dimension that differed from a natural number. That is weird: Hey, this is a 2.4-dimensional object 😉

There are lots of higher-dimension fractals. The Menger sponge is pretty basic, but the quaternion Julia fractal is 4-dimensional and totally hot (usually rendered as a 3-d slice of the 4-d set). It looks like taffy!

Why are mathematical shapes beautiful? Symmetry is one big factor in that, although definitely not the only factor.

And this is seven-fold symmetry, like Sili said. You don’t see that too often. I would be interested to know if there are any biological examples of that, though. There are sea stars which appear to have five-fold symmetry, but then when you see sea stars with lots more legs, you realize it’s just rotational symmetry and the “5” or “12” is just a coincidence.

More interesting than the fact of the rendering (which is indeed beautiful, and I’ve raved at my own blog about it previously) is that the artists found something worth rendering. The idea of a three-dimensional Mandlebrot set that is as rich as the well known two-dimensional one has been around pretty much as long as the set has, but it’s only now that someone has found what looks like the real thing — a 3D analogue. This is not a triumph of technology (nifty rendering of complex math — something we already do trivially these days) but a triumph of mathematical discovery. This structure is something new with properties worth rendering.

Just scroll down to the end of the post to get the link to the program! It is Windows only now and requires a 3D card that can handle Shader Model 3. Linux and Mac versions are on their way and should be done tomorrow or so!

Why are mathematical shapes beautiful? I think the answer is that some are and some aren’t. The ones that are attractive tend to get looked at for longer, shared more often and remembered more easily. The ones that aren’t so visually appealing just get remembered as tools, or results without anyone thinking “that wasn’t very good looking.”

One possible explanation involves the evolution of human sexual selection. Study after study has shown the humans prefer (unconsciously) facially and bodily symmetrical choices in mates. The brain devotes a great deal of processing power to detect the slightest deviations. The adaptive explanation is that less symmetrical individuals tended to be less healthy and thus bore fewer offspring; thus the dominant preference among survivors would be for symmetry.

The love of mathematical symmetry and elegance probably is a by-product of those cognitive adaptations for mating.

“Carmelized Hazelnut Swirl?” No. No, it’s Yog-Sothoth. The gaping mouths and heavy-lidded eyes are a giveaway. I’m sorry, but it could not be more obvious.

“Ice Cream From Neptune?” Again, no. They are alien, misshapen skulls out-stretched on tortured necks, craning for the sun from an airless prison at the bottom of a crevasse. Either that or they’re ponies.

The theory that I have (and it’s mine) is this: ehhhh-hem. (this is it (not that, obviously, but what comes next))

Multidimensional mathematical constructs are beautiful because they mimic the “shape” of our thoughts, the patterns of neural firings inside our brains that exist not only in three physical dimensions but through the dimension of time. We all tend to like things that remind us of ourselves, and thus our minds are subconsciously responding to something which begins to mimic itself.

For some reason, when I see that 3D mandelbrot, I see a whole bunch of sarlaccs on a sarlacc.
I see a really intricate crocheted tea-cosy…..
Seriously, if I posted this pic to one of the LJ knitting/crochet comms, you can bet that somebody will try to figure out the pattern and reproduce it.

Creationists say that complexity is evidence of intelligent design, fractals are an excellent example of why this is poor reasoning. There is no intelligence behind what numbers do when combined in simple formulas, and yet the result can be infinitely complex.

Real fractals only work in even dimensions due to quaternion properties.

That is not what the article that BA refers to claims, I think.

Seems to me that escape-time fractals, used to get among other fractals the Mandelbrot set, may be based on even dimensioned vector fields.

But for example the iterated function system fractals, such as the Sierpinski gasket, seems to admit any whole dimension as base. See the illustration of “A Sierpinski square-based pyramid and its ‘inverse'” in Wikipedia.

[And likely you can “fractalize” a fractal, in which case you can iterate based on any Hausdorff dimension.]

I’ve seen shapes like these when tripping. I expect the mathematical laws that determine how a structure unfolds from a simple DNA program are of this type, though I feel we have a way to go before we have the basic formula for a true multi-dimensional fractal. If you were to wander thru the synaptic connections in your brain, I suspect they would look a lot like this.
(I’m actually referring to “When Hell Froze Over”).

Then I wonder how the false vacuum would look, with all its twisty/windy micro-worm holes.

Torbjörn Larsson is right – fractals can happen in any dimension. The Mandelbrot set and its relatives can only happen in even dimensions because they require a field of complex numbers, which don’t work right in three dimensions. The set in this render is based off of a 3-dimensional analog of the complex numbers that share some but not all of the properties of true complex numbers.

Phil, one really great thing about Pascal was that its function definitions could themselves contain function definitions, i.e. they were fractal in a sense where one could build hierarchies of rules for each level of an application.

Now suppose space-time is fractal, so that (somewhat) different laws apply at each level of space or time detail. In other words, at the galaxy level, F=MA may be a tad different, and thus accommodate the dark matter effect. Newton discovered F=MA for the level where we live; Einstein discovered the relativistic level, where space-time gets adjusted. And Planck hit on the quantum level where (perhaps) local time reversal is possible and evolutionary mutations might take place as quantum jumps.

(I have discovered a marvelous proof of sll this, but the comment box is too small to contain it.)

I doubt it’s so much different laws at each level as it is different ways of looking at the same underlying reality. Kinda, sorta like this:

A STATIC electron appears to us to produce an electric field(a unidirectional effect that decreases as the inverse square of the distance from the electron)
A MOVING electron appears to produce an electromagnetic field(a bi-polar effect that also decreases as the inverse square)
An ACCELERATED electron(ie, changing it’s velocity) appears to produce an electromagnetic wave, which has no poles, neither is it a field but it still decreases as the inverse square.

About the only thing all these effects have in common is that they’re all produced by an electron(ok, ANY charged particle) and the effect decreases as the inverse square. We just see the effects from different points of view.

I expect the universe behaves pretty much the same way, regardless of the scale.

If the LHC succeeds in making micro black holes, it will be precisely because gravity deviates from an inverse square law at small scales. And if it does, it will be because, on those tiny scales, spacetime has more than three spatial dimensions (inverse square laws are a natural consequence of three spatial dimensions, BTW). They’re just too tiny (“rolled up”) for us to experience in the macro world.

The well-known Mandelbrot set isn’t a 2-dimenional set, it’s actually a 2-dimensional slice of a 4-dimensional set. (Its orthogonal analog is the Julia set, which the Mandelbrot set was originally devised as a means to catalog the Julia set.) We just happen to usually look at the Mandelbrot slice where Z0=0+0i. You get some pretty neat results by modifying that.

Coastlines are fractal in the sense that they continue to look like coastlines no matter how closely you examine them, even as you end up trying to trace water molecules jaggedly butting up against sand molecules.

For you programmers, to draw fractal lines, define a recursive function that draws a line by first offsetting its midpoint sideways a little, then drawing the two halves of the line by calling the same function. This will draw all straight lines using tiny curved increments. CAUTION: To avoid system lockup due to infinite recursion, add a check that skips drawing the line for very short lines, or draws the given line without moving its midpoint.

Maybe I just watch/read too much science fiction… but Phil’s comment about how some forms of life, like Brocholi, are 3-d fractals… just makes me wonder if there are any intelligent life forms out there which are also fractal based…

Imagine a Dr. Who story where the TARDIS gets infected by such a creature, and it begins re-arranging the internal geometry of the space/time ship to suit itself…

The original article briefly touches on an interesting rendering problem with the 3-D version. To get the 3-D appearance, you have to do lighting calculations. But these depend on the surface tangents, and those are undefined on the fractal! So you always have to stop at some chosen depth, and render the approximation. Remember that as you look at each image – there’s an infinite amount of unresolved detail at every point.

Fractals, I love ’em. Each one is distinct place in a mathematical landscape. And each of those places is:

FULL OF PLACES!!

Which are full of places and it just won’t quit. I find that so delightfully baffling; just knowing it.

I generated my first fractals on an IBM clone, 8088 processor running at 4.7Mhz. A full screen 640x480x256 VGA rendering took about a day and a half with the iteration variable turned down low. I was ecstatic! I even did some BASIC coding to render using the L-System to make ferns and gaskets and all. Was that really almost 25 years ago?

Coastlines are fractal in the sense that they continue to look like coastlines no matter how closely you examine them, even as you end up trying to trace water molecules jaggedly butting up against sand molecules.

I love fractals. Back in high school I wrote a Mandelbrot fractal generator for my Casio calculator. Later I extended it to use all the four colours the calculator screen offers. Rendering a frame took over an hour!

I used to explore fractals in the excellent XaoS fractal zoomer along with my roomate. When we found a beautiful area of the fractal, we plugged the coordinates into my calculator, gave it the night to render, and woke up to a less beautiful calculator screen, though still charming.